W. G. Dwyer

Function complexes in homotopical algebra

1 .l Summary IN [l] QUILLEN introduced the notion of a model category (a category together with three classes of maps: weak equivalences, fibrations and cofibrations, satisfying certain axioms (1.4 (iv))) as a general framework for “doing homotopy theory”. To each model category M there is associated a homotopy category. If W C M denotes the subcategory of the weak equivalences, then this homotopy category is just the localization M[W-‘I, i.e. the category obtained from M by formally inverting the maps of W, and it thus depends only on the weak equivalences and not on the fibrations and the cofibrations. Moreover, if two model categories are connected by a pair of adjoint functors satisfying certain conditions, then their homotopy categories are equivalent. The homotopy category of a model category M does not capture the “higher order information” implicit in M. In the pointed case, however, Quillen was able to recover some of this information by adding some further structure (a loop functor, a suspension functor and fibration and cofibration sequences) to the homotopy category. His fundamental comparison theorem then stated that, if two pointed model categories are connected by a pair of adjoint functors satisfying certain conditions, then their homotopy categories are equivalent in a manner which respects this additional structure. The aim of the present paper is to go back to an arbitrary model category M and construct a simplicial homotopy category which does capture the “higher order information” implicit in M. This simplicial homotopy category is defined as the hummock localization L”(M, W) (for short LHM) of [2]. It is a simplicial category (1.4) with the following basic properties: (i) The simplicial homotopy category LHM depends (by definition) only on the weak equivalences and not on the fibrations and cofibrations. (ii) If two model categories are connected by a pair of adjoint functors satisfying Quillen’s conditions, then their simplicial homotopy categories are weakly equivalent (1.4). (iii) The “category of components” of the simplicial homotopy category of M is just the homotopy category of M. (iv) If M, is a closed simplicial model category [I], then, as one would expect, the full simplicial subcategory M

Calculating simplicial localizations

C M* generated by the objects which are both cofibrant and jibrant is weakly equivalent (1.4) to LHM. (v) “LHM provides M with function complexes”, i.e. for every two objects X, YE M, the simplicial set LHM(X, Y) has the correct homotopy type for a function complex, in the sense that, for every cosimplicial resolution X* of X and every simplicial resolution Y, of Y (4.31, it has the same homotopy type as diag M(X*, Y*).

Homotopy fixed-point methods for Lie groups and finite loop spaces

1.1. Summary. This paper is essentially a continuation of [3], where we introduced a (standard) simplicial localization functor, which assigned to every category C and subcategory W c C, a simpficiaf category LC with in each dimension the same objects as C (i.e. for every two objects X, YE C, the maps X -+ YE LC form a simplicial set LC(X, Y)). This simplicial localization has all kinds of nice general properties, but, except in a few extreme cases [3, Section 51, it is difficult to get a hold on the homotopy type of the simplicial sets LC(X, Y). In this paper we therefore consider a homotopy variation on the standard simplicial localization LC, the hammock localization LHC (Section 2), which (Section 3) has some of the nice properties of the standard localization only up to homotopy, but is in other respects considerably better behaved. In particular (Sections 4 and 5) the simplicial sets LHC(X, Y) are much more accessible; each simplicialsetLHC(X, Y) is the direct limit of a diagram of simplicial sets which are nerves ofcategories and (Section 6) if the pair (C, W) admits a “homotopy calculus of fractions,” then several of these nerves already have the homotopy type of LHC(X, Y). When W satisfies a mild closure condition this happens, for instance, if (Section 7) the pair (C, W) admits a calculus of feft fractions in the sense of Gabriel-Zisman [5] or if (Section 8) W is closed under push outs, in which case LHC(X, Y) has the homotopy type of the nerve of the category which has as objects the sequences X + C t Y in C for which the second map is in W and which has as maps the commutative diagrams

Homotopy limit functors on model categories and homotopical categories

A loop space X is by definition a triple (X, BX, e) in which X is a space, BX is a connected pointed space, and e: X -QBX is a homotopy equivalence from X to the space QBX of based loops in BX. We will say that a loop space X is finite if the integral homology H*(X, Z) is finitely generated as a graded abelian group, i.e., if X appears at least homologically to be a finite complex. In this paper we prove the following theorem.

Homotopy commutative diagrams and their realizations

Model categories: An overview Model categories and their homotopy categories Quillen functors Homotopical cocompleteness and completeness of model categories Homotopical categories: Summary of part II Homotopical categories and homotopical functors Deformable functors and their approximations Homotopy colimit and limit functors and homotopical ones Index Bibliography.

A classification theorem for diagrams of simplicial sets

Abstract In this paper we describe an obstruction theory for the problem of taking a commutative diagram in the homotopy category of topological spaces and lifting it to an actual commutative diagram of spaces. This directly generalizes the work of G. Cooke on extending a homotopy action of a group G to a topological action of G.

Homotopy theory and simplicial groupoids

1.1 Summary THE AIM of this paper is to prove a rather general classification theorem for diagrams of simplicial sets, which encompasses the classification results for Postnikov conjugates of [15] and [3] and those for simplicialfibrations of [1] and [4]. This theorem will be applied in [10] to analyze the category of topological spaces on which a topological group G acts. It leads to a classification of these G-spaces with respect to weak equivariant homotopy equivalences, i.e. with respect to equivariant maps which restrict to weak homotopy equivalences on the fixed point sets of a given collection of subgroups of G. 1.2 Motivation To motivate our result we start with recalling the essence of the two classification results mentioned above:

Singular functors and realization functors

The homotopy theory of simplical groups is well known [2, Ch. VI] to be equivalent to the pointed homotopy theory of reduced (i.e. only one vertex) simplicial sets (by means of a pair of adjoint functors G and W. The aim of this note is to show that similarly, the homotopy theory of simplical groupoids is equivalent to the (unpointed) homotopy theory of (all) simplical sets. This we do by 1. (i) showing that the category of simplicial groupoids admits a closed model catagory structure in the sense of Quillen [3], and 2. (ii) extending the functors G and W to pair of adjoint functors G: (simplicial sets)↔(simplicial groupoids): W which induce the desired equivalence of homotopy theories. We also show that the category of simplical groupoids admits a simplical structure which produces “function complexes” and “simplical monoids of self homotopy equivalences” of the correct homotopy types.

Homotopical uniqueness of classifying spaces

Abstract In [6] Quillen showed that the singular functor and the realization functor have certain properties which imply the equivalence of the weak homotopy theory of topological spaces with the homotopy theory of simplicial sets. The aim of this note is to generalize this result and to show that one can, in essentially the same manner, establish the equivalence of other homotopy theories (e.g., the equivariant homotopy theories) with homotopy theories of simplicial diagrams of simplicial sets. Applications to equivariant homotopy will be given in [3] and [4].

Function complexes for diagrams of simplicial sets

If G is a connected compact Lie group, then for almost all prime numbers p the mod p cohomology ring of the classifying space BG is a finitely generated polynomial algebra. In 1961, N. Steenrod [24] asked in general for a determination of all spaces X such that H∗(X,Fp) is a finitely generated polynomial algebra (i.e., such that X has a polynomial cohomology ring); at that time, the only examples known were spaces of the form X = BG. There has been a lot of subsequent progress on this problem. On one hand, the topological constructions of Sullivan [25, p. 4.28], as exploited by Clark-Ewing [8] and Wilkerson [27], have led to the discovery of exotic spaces X with polynomial cohomology rings. On the other hand, the algebraic arguments of Wilkerson [26] and Adams-Wilkerson [1] have shown in some generality that if X is any space with a polynomial cohomology ring then H∗(X,Fp) must be one of the polynomial algebras listed in [8]. However, there is still a gap here between topology and algebra; in this paper we act to narrow the gap and in some cases to close it. Suppose that p is an odd prime. Let K be the category of unstable algebras [18] over the mod p Steenrod algebra Ap and Kpoly the full subcategory of K consisting of objects which as rings are finitely generated polynomial algebras. If X is a p-complete space with H(X,Fp) ∈ Kpoly we extend the ideas of [1] by associating to X a finite p-adic linear group WX generated by pseudoreflections (see 1.1); given any such finite group W such that p does not divide the order of W, we then show (1.2) that there is up to homotopy exactly one p-complete space X with H∗(X,Fp) ∈ Kpoly and WX = W. In a variety of particular situations (1.3, 1.4) this gives a bijective correspondence between finite group data and homotopy types of p-complete spaces with polynomial cohomology rings.